Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
semilinear parabolic equation; functional differential equation; integrodifferential equation; integral equation fractional evolution equation; global existence; stability; variation of parameters
semilinear parabolic equation; functional differential equation; integrodifferential equation; integral equation fractional evolution equation; global existence; stability; variation of parameters
Summary:
We prove global existence and stability results for a semilinear parabolic equation, a semilinear functional equation and a semilinear integral equation using an inequality which may be viewed as a nonlinear singular version of the well known Gronwall and Bihari inequalities.
References:
[1] G. Butler, T. Rogers: A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations. J. Math. Anal. and Appl. 33 No 1 (1971), 77–81. MR 0270089 | Zbl 0209.42503
[2] G. DaPrato, M. Iannelli: Regularity of solutions of a class of linear integrodifferential equations in Banach spaces. J. Integral Equations Appl. 8 (1985), 27–40. MR 0771750
[3] W. E. Fitzgibbon: Semilinear functional differential equations in Banach space. J. Diff. Eq. 29 (1978), 1–14. MR 0492663 | Zbl 0392.34041
[4] A. Friedman: Partial Differential Equations. Holt, Rinehart and Winston, New York, 1969. MR 0445088 | Zbl 0224.35002
[5] Y. Fujita: Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27 (1990), 309–321. MR 1066629 | Zbl 0796.45010
[6] H. Hattori, J. H. Lightbourne: Global existence and blow up for a semilinear integral equation. J. Integral Equations Appl. V2, No4 (1990), 529–546. MR 1094482
[7] D. Henry: Geometric theory of semilinear parabolic equations. Springer-Verlag, Berlin, Heidelberg, New York, 1981. MR 0610244 | Zbl 0456.35001
[8] H. Hoshino: On the convergence properties of global solutions for some reaction-diffusion systems under Neumann boundary conditions. Diff. and Int. Eq. V9 No4 (1996), 761–778. MR 1401436 | Zbl 0852.35023
[9] M. Kirane, N. Tatar: Asymptotic stability and blow up for a fractional evolution equation. submitted.
[10] M. Medved’: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. and Appl. 214 (1997), 349–366. MR 1475574 | Zbl 0893.26006
[11] M. Medved’: Singular integral inequalities and stability of semilinear parabolic equations. Archivum Mathematicum (Brno) Tomus 24 (1998), 183–190. MR 1629697 | Zbl 0915.34057
[12] M. W. Michalski: Derivatives of noninteger order and their applications. ”Dissertationes Mathematicae”, Polska Akademia Nauk, Instytut Matematyczny, Warszawa 1993. MR 1247113 | Zbl 0880.26007
[13] M. Miklavčič: Stability for semilinear equations with noninvertible linear operator. Pacific J. Math. 1, 118 (1985), 199–214. MR 0783024
[14] S. M. Rankin: Existence and asymptotic behavior of a functional differential equation in a Banach space. J. Math. Anal. Appl. 88 (1982), 531–542. MR 0667076
[15] R. Redlinger: On the asymptotic behavior of a semilinear functional differential equation in Banach space. J. Math. Anal. Appl. 112 (1985), 371–377. MR 0813604 | Zbl 0598.34053
[16] C. Travis, G. Webb: Existence and stability for partial functional differential equations. Trans. Amer. Math. Soc. 200 (1974), 395–418. MR 0382808 | Zbl 0299.35085
[17] C. Travis, G. Webb: Existence, stability and compacteness in the $\alpha $-norm for partial functional differential equations. Trans. Amer. Math. Soc. 240 (1978), 129–143. MR 0499583
Search
Partner of
